Mathematical Physics

In the first part of my talk I'll briefly review some aspects of the relations between N=4, d=4 SYM and vertex operator algebras (VOAs) discussed in recent work of Gaiotto and collaborators. The resulting picture predicts conjectural generalisations of the geometric Langlands correspondence. We will focus on a class of examples figuring prominently in recent work of Creutzig-Dimofte-Garner-Geer, labelled by parameters n (rank) and k. For the case k=1,n=2 we will point out that the conformal blocks of the relevant VOA, twisted by local systems, represent sections of natural holomorphic line bundles over the moduli spaces of local systems closely related to the isomonodromic tau functions. Observing the crucial role of (quantised) cluster algebras in the definition of the holomorphic line bundles suggests natural generalisations of this story to higher values of the parameters k and n.

I will give an introduction to 4d N=2 class-S theories. I will describe the construction of such theories, the roles played by extended defects such as line defects and surface defects, as well as connections to Hitchin systems.